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Difficulty: EasyFundamental Trigonometric Identities

An angle θ\theta lies in the second quadrant and satisfies sin⁡θ=513\sin\theta = \frac{5}{13}. What is the value of 12tan⁡θ12\tan\theta?

Answer: -5

Answer

The value of 12tan⁡θ12\tan\theta is -5.
Applying the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 with sin⁡θ=513\sin\theta = \frac{5}{13} yields cos⁡2θ=144169\cos^2\theta = \frac{144}{169}. Since the angle θ\theta lies in the second quadrant, its cosine is negative, meaning cos⁡θ=−1213\cos\theta = -\frac{12}{13}. Using the quotient identity tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, we find tan⁡θ=−512\tan\theta = -\frac{5}{12}. Multiplying by 12 gives the correct value of -5.

Step-by-Step Solution

1
Use the Pythagorean identity to find the magnitude of cos⁡θ\cos\theta.
cos⁡2θ=144169\cos^2\theta = \frac{144}{169}
The Pythagorean identity states that sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Substituting sin⁡θ=513\sin\theta = \frac{5}{13} gives cos⁡2θ=1−(513)2\cos^2\theta = 1 - \left(\frac{5}{13}\right)^2.
2
Determine the value of cos⁡θ\cos\theta by applying the quadrant sign rule.
cos⁡θ=−1213\cos\theta = -\frac{12}{13}
Since θ\theta is in the second quadrant, the cosine of θ\theta must be negative.
3
Calculate tan⁡θ\tan\theta using the quotient identity.
tan⁡θ=−512\tan\theta = -\frac{5}{12}
The quotient identity is tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}.
4
Multiply tan⁡θ\tan\theta by 12 to find the required expression's value.
-5
Multiplying the value of tan⁡θ\tan\theta (which is −512-\frac{5}{12}) by 12 yields −5-5.

Key Concept

Fundamental Trigonometric Identities
Estimated Time:1m 0s
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Topics

Fundamental Trigonometric Identities

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